
where \( {\mathbf {R}}_{\mathrm {BS},g}=\beta _{\mathrm {d},g_{k}}{\mathbf {R}}_{\mathrm {BS},g_{k}} \in \mathbb {C}^{M \times M} \)  and \( {\mathbf {R}}_{\mathrm {IRS},g}=\beta _{2,g_{k}}{\mathbf {R}}_{\mathrm {IRS},g_{k}} \in \mathbb {C}^{N \times N} \)  represent the deterministic Hermitian-symmetric positive semi-definite aggregate covariance
matrices at the BS and the IRS respectively corresponding to group \( g \)  with \( \left({\mathbf {R}}_{\mathrm {BS},g} \right)=M \)  and \( \left({\mathbf {R}}_{\mathrm {IRS},g} \right)=N \) . Note that \( \beta _{\mathrm {d},g_{k}} \) , \( {\mathbf {R}}_{\mathrm {BS},g_{k}} \)  and \( \beta _{2,g_{k}}\) , \( {\mathbf {R}}_{\mathrm {IRS},g_{k}} \)  are the path-losses, correlations of the BS-UE \( k \)  and IRS-UE \( k \)  links in group \( g \) , respectively. This channel modeling is very versatile since it can account for both correlation (\({\mathbf {R}}_{\mathrm {BS},g_{k}} ,{\mathbf {R}}_{\mathrm {IRS},g_{k}}\) ) and path loss (\( \beta _{\mathrm {d},g_{k}},\beta _{2,g_{k}} \) ) simultaneously or either of the two. Certain estimation methods (see e.g., [1]}, [2]}) allow to obtain the correlation matrices and the path-losses,
which can, thus, be assumed known by the network. Moreover, \( {\mathbf {z}}_{\mathrm {d},g_{k}} \sim \mathcal {CN}\left({\mathbf {0}},{\mathbf {\mathrm {I}}}_{M}\right) \)  and \( {\mathbf {z}}_{2,g_{k}}\sim \mathcal {CN}\left({\mathbf {0}},{\mathbf {\mathrm {I}}}_{N}\right) \)  express the corresponding fast-fading vectors.
